Munich Personal RePEc Archive
Systemic Value Added: an alternative to
EVA as a residual income model
Magni, Carlo Alberto
University of Modena and Reggio Emilia, Modena, Italy
January 2001
Online at https://mpra.ub.uni-muenchen.de/7763/
MPRA Paper No. 7763, posted 15 Mar 2008 17:09 UTC
1
Systemic Value Added: an alternative to EVA
as a residual income model
Carlo Alberto Magni
Department of Economics, Faculty of Economics
University of Modena and Reggio Emilia
Email: [email protected]
Abstract. This work presents a notion of residual income called Systemic Value Added
(SVA). It is antithetic to Stewart’s (1991) EVA, though it is consistent with it in overall
terms: a project’s Net Final Value (NFV) can be computed as the sum of capitalized EVAs or
as the sum of uncapitalized SVAs. As a result, SVA and EVA decompose the NFV in
different ways. Two numerical examples show the application of the model proposed. The
two notions are the result of a different cognitive approach. The existence of possible formal
translations of the residual income concept induces to regard residual income as a mere
conventional notion.
Foreword. This is the English translation of the following paper:
Magni, C. A. (2001). Valore aggiunto sistemico: un’alternativa all’EVA quale indice di sovraprofitto periodale, Budget, 25(1), 63−71. The original paper is reproduced after the English version
2
1. Introduction
Stewart’s (1991) Economic Value Added is a formal translation of the classical notion of
residual income. It is used for valuing a firm or a project or for management compensation
(Biddle, Bowen and Wallace, 1999). This paper presents an approach which is alternative to
the standard notion of residual income implied by EVA; the approach is based on a different
interpretation of the notion of “residual income” and two simple numerical examples are
presented for clarifying purposes. The model proposed, which is here called Systemic Value
Added, is based on a systemic notion of residual income, where the diachronic evolution of the
investor’s financial system is relevant. The standard residual income, of which EVA is one
instantiation, is contrasted with the SVA approach, and analogies and differences will be
considered, both for unlevered and levered projects (the latter case is, methodologically, only
a simple generalization). The SVA approach has applicative implications because it provides
residual income measures which the standard models such as EVA are not capable of
individuating. The two models offer different information, though each of them can be said to
be a residual income model. The choice of either model is conventional.
2. EVA
The Economic Value Added for an n -period project (firm) in the s -th period is computed
as
1-se1-s1-s
e1-s1-ss
s
1-ssss
IC)V D
V D ROI(
IC )WACC (ROIEVA
∗+
∗+∗−=
∗−=
iδ (1)
s =1,2,…, n . sWACC is the (Weighted Average Cost of Capital), sδ is the cost of debt,
sROI is the return on investment, 1-sIC is the capital invested at the beginning of the period,
1-sD is the value of debt, 1-sV is the value of equity, i is the equity cost of capital.
3
Henceforth, it will be assumed that the project is unlevered (zero debt). This assumption is
made for mere expositional convenience and will be relaxed in the second numerical example
in section 6.1 With zero-debt assumption, eq. (1) may be written as
1-sss IC) (ROIEVA ∗−= i (2)
where i is the cost of capital. The n-period aggregate residual income, defined Market Value
Added (MVA), is found by summing the EVAs , previously discounted at a rate 'i :
s-
1s ) '(1EVAMVA i
n
s
+=∑=
(3a)
In principle, one could refer the MVA to time n , so that
s-
1s ) '(1EVAMVA n
n
s
i+=∑=
(3b)
If ii =' it is easy to show that eqs. (3a) and (3b) coincide with the project’s Net Present Value
(NPV) and Net Final Value (NFV), respectively (Esposito, 1998; Magni, 2000a) and that
Stewart’s model is equivalent to the NPV (NFV) decomposition model by Peccati (see Magni
2000a, 2000b).
3. SVA
The EVA approach has proved a success in most recent years, and it seems that eq. (2) is a
natural formal translation of the notion of residual income (excess profit). In fact, it is only
one possible interpretation. An alternative representation of the economic notion of residual
income (also known as excess profit). is the following: suppose the decision maker has the
1 It is worth stressing that such an assumption is irrelevant because the differences between the two models pertain to alternative interpretations of the notion of residual income. As we will focus on the cognitive perspective, to deal with unlevered projects makes description simpler while shedding lights on the relevant features of the problem.
4
opportunity to invest in an economic activity, say P, consisting of a sequence of cash flows
sa R∈ , s =0,1,…, n and let x be the return rate of the investment (assumed constant). Basic
notions of financial mathematics tell us that the capital invested in the operation at the
beginning of each period is
00IC a−=
ss ax −+= )1(ICIC 1-s s =1,2,…, n
which implies
ks
k
ks xax−
=+−= ∑ )1()(IC
s
0
s =1, 2,…, n (4)
where the dependence of sIC on the return rate is highlighted. The invested capital is
therefore expressed as the compounded sum at time s , calculated at the rate x , of the first
s +1 cash flows. Obviously, we have 0IC =n because x is the internal rate of return.
Consider now the quantity obtained from eq. (4) by replacing the rate x with the
opportunity cost of capital i . We have
ks
k
ks iai −
=+−= ∑ )1()(IC
s
0
s =1,2,…, n . (5)
Computing the EVA of this investment by making use of eq. (2) we get
)(IC -)(IC EVA 1-s1-ss xixx ∗∗= (6)
The proposal alternative to EVA boils down to employing eq. (6) where the term
)(IC 1-s xi ∗−
is replaced by
)(IC 1-s ii ∗− .
So doing, we obtain what is here called the Systemic Value Added (SVA):
5
)(IC -)(IC SVA 1-s1-ss iixx ∗∗= (7)
4. The different meanings of the residual income notion
The passage from eq. (6) to eq. (7) is delicate, because the substitution of the internal rate
of return with the cost of capital has major consequences in terms of interpretation. To grasp
the economic-financial meaning of eqs. (6)-(7) let us focus on the decision process. Suppose
that the initial decision maker’s wealth is RE ∈0 . Suppose also that she can borrow and lend
funds at the cost of capital i . This means that every positive (negative) cash flow generates
positive (negative) interest at a rate i and that at time 0 the investor renounces to investing
0IC at the rate i and invests it in the economic activity P. The investor’s wealth sE at time s
is
])1()1[()1((4)]by [
)1()1()(IC
00
00
kskss
k
ks
kss
k
ks
ss
xiaiE
iaiExE
−−
=
−
=
+−+++==
++++=
∑
∑ (8)
Eq. (8) may be explained through an “accounting” representation of the investor’s financial
system:
Applications Sources
sss aiSS ++= − )1(1 | 111
1111
)(IC
)(IC)(IC
−−−
−−−−
++=+++=
sss
sssss
iSxxE
iSxxSxE
sss axxx −+= − )1()(IC)(IC 1 |
(9)
where 000 aES += and, obviously, )(IC xSE sss += . Such a representation describes the
diachronic evolution of the investor’s financial system, which is structured in a portfolio of
6
two investments, activity P and an asset which we can call account S. Their values are,
respectively, )(IC xs and sS .2 The profit from this portfolio is
111 )(CI −−− +=− ssss iSxxEE .
On the other side, in case of rejection of P, the initial wealth would have been invested in
account S at the rate i , and the investor’s wealth at time s , say sE , would have been
ssiEE )1(0 +=
whence the profit
111 −−− ==− ssssiSiEEE
because
Applications Sources
)1(1iSS
ss += − | 11 −− += sssiSEE
(10)
with 00
ES = . Hence, once computed the two profits relative to the alternative situations
(accept/reject P), the difference between them may be interpreted as a residual income, i.e. as
the profit from the ‘accept’ alternative over and above the profit from the ‘reject’ alternative.
It is, so to say, the value that is added to that profit that could be achieved by investing at the
rate i . The value added is here labelled systemic because it is drawn from considerations
about the evolution of the investor’s financial system:
2 Given that it is often 00 <a , the first cash flow is a withdrawal from account S, which “finances” P, so to say,
at a cost of i (the financing is a virtual one if 00 >S , in the sense of investment’s lost opportunity).
7
111
11s )(IC)()(SVA −
−−−
− −+=−−−= sss
ssss iSiSxxEEEE (11a)
or
111
1s
11 )(IC)(SVA)( −
−−−−
− −++−=+−=− sss
ssssss iSiSxxEEEEEE . (11b)
Eq. (11) may be rewritten as
which proves coincidence with eq. (7).
In this way, the notion of excess profit implied by the SVA model refers to a comparison
between profits concerning two different financial systems, pertaining to different courses of
action. Investment P presupposes investment of )(IC xs at the return rate x , whereas the
alternative course of action is represented by the investment of )(IC is at the rate i . The
difference measures the residual income.
Conversely, the classical idea of residual income summarized in the EVA equation stems
from the following line of reasoning: at the beginning of each period the investor has the
opportunity of investing the amount )(IC xs at the rate x in activity P or, alternatively,
investing the same amount at the rate i in account S. The residual income is given by the
comparison between these two alternatives, whence eq. (6).
The two models conciliate at an aggregate level. As anticipated in eq. (3b), the sum of
compounded EVAs coincides with the NFV; the latter is also obtained as the uncompounded
sum of the SVAs: from eq. (11) we get
)(IC)(IC
)1()(IC
)()(CISVA
11
01
11
1s
iixx
iaixx
SSixx
ss
kss
k
ss
ss
s
−−
−
=−
−−
−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−=
−−=
∑
8
This means that the two models decompose NFV in different periodic shares, though being
consistent in overall terms. The meaning of this conciliation is enlightening: if the time
interval considered is the entire span of n periods, the aggregate residual income is the Net
Final Value (or Net Present Value, if it is referred to initial date). If one decomposes such an
aggregate excess profit into periodic shares, the process of imputations contains unavoidable
conventional elements. The two interpretations stem from two alternative cognitive
perspectives. The SVA model and the EVA model show that the idea of excess profit
(residual income) is not univocal, and that different formal translations can be legitimately
considered translations of the same concept. Borrowing terminology from Duhem (1914), one
may well claim that to a determined practical fact there corresponds multiple theoretical facts.
Actually, the practical fact is not so “practical”: it consists of a comparison between two
alternative courses of action, and a comparison is always a mental fact, whose content
depends on the outlook followed in its description. Residual income is not cash, it is (or,
better, it derives from) counterfactual conditionals such as “if it were not…then it would
be…” or “if it had not been…then it would have been…”. They measure the “how more” or
“how less” with respect to an alternative that could be or could have been. This induces to
think that the idea of residual income is an intrinsically conventional mental fact and that the
choice of which one should be the formal translation to be employed depends on the piece of
information the decision maker is willing to obtain.
5. Numerical example (zero-debt assumption)
Following are two simple numerical examples aimed at familiarizing readers with the
SVA model and better understand the differences from Stewart’s EVA.
NFV)1(
)1()1()1(
)()(SVA
0
00
0
1
11
1ss
=+=
+−+++=
−=−−−=
−
=
−
=
=
−−
=
∑
∑
∑∑
snn
s
s
nsnn
s
sn
nn
n
s
ssss
n
ia
iEiaiE
EEEEEE
9
Suppose an investor has the opportunity to invest in a project A whose cash flows are
10000 −=a 6001 =a 4502 =a 1103 =a at time 0, 1, 2, 3 respectively. Graphically, we may
represent the project as:
0 1 2 3
|---------------------------------|----------------------------------|----------------------------------|
-1000 600 450 110
Assume the project is unlevered, the initial investor’s wealth is 15000 =E and the
opportunity cost of capital is 09.0=i , the NPV and the NFV of A are, respectively,
155.14)09.01(110)09.01(450)09.01(6001000NPV 321 =++++++−= −−−
Keeping eyes on eqs. (6), (7), (9) and (11) and noting that A’s internal return rate is 1.0=x ,
we have
331.18)09.01(VAN110)09.01(450)09.01(600)09.01(1000NFV 323 =+=++++++−=
8745.1960110)09.01(05,1698
05,1698450)09.01(1145
1145600)09.01(500
50010001500
3
2
1
0
=++==++=
=++==−=
S
S
S
S
5435.1942)09.01(15.1782
15.1782)09.01(1635
1635)09.01(1500
1500
3
2
1
0
=+=
=+=
=+=
=
S
S
S
S
10
1000)(IC 00
0 =−= SSi 1000)(IC0 =x
490)(IC 11
1 =−= SSi 500600)1.01(1000)(IC1 =−+=x
1.84)(IC 22
2 =−= SSi 100450)1.01(500)(IC2 =−+=x
NFV331.18)(IC 33
3 −=−=−= SSi 0110)1.01(100)(IC3 =−+=x
whence
This example sheds light on the conventions used fo interpreting the notion of residual
income. Just focus on the second residual income. “Mister EVA” reasons as follows:
“500 is the capital to be invested at the beginning of the second period. If I invest it at
a rate of 10% I get an income of 50; if, instead, I invest it at a rate of 9% I get 45. The
difference is 5, that is, to invest in A in the second period means that to get a residual
income equal to 5.”
Conversely, “Mister SVA” reasons as follows:
“If today I choose to invest in A, the capital invested in the project at the beginning of
the second period will be 500, from which I get a 10% return rate, which entails an
income of 50. But, so doing, the value of account S will be, at the beginning of the
second period, smaller than it would be if today I invested my funds at the rate 9%; in
particular, it will be smaller by an amount of 490. As a result, this investment implies a
foregone return equal to 44.1 (=0.09*490). The residual income is therefore 5.9 (= 50
– 44.1).”
110009.01001.0EVA
550009.05001.0EVA
10100009.010001.0EVA
3
2
1
=∗−∗==∗−∗==∗−∗=
431.21.8409.01001.0SVA
9.549009.05001.0SVA
10100009.010001.0SVA
3
2
1
=∗−∗==∗−∗==∗−∗=
NFV331.181)09.1(5)09.1(10EVA)1(EVA)1(EVA 232
21 ==++=++++ ii
NFV331.18431.29.510SVASVASVA 321 ==++=++
11
The two lines of reasoning are different, but they both measure residual income in the second
period. The fact is that the notion of residual income (excess profit) are ambiguous, for it is
possible to rest on two different lines of reasoning, the choice between the two being
conventional. To Mister EVA the alternative course of action is the investment of )(IC xs at
the rate i , whereas to Mister SVA the alternative course of action is the investment of )(IC is
at the rate i . Which is the best one? It depends on the pieces of information one is willing to
draw. Only the decision maker knows which is the approach best suited to her own needs.
Certainly, the conventional elements suggest caution in the indiscrimate use of the EVA as
performance index or as a basis for compensation plans. EVA is only one possible approach
to performance valuation, not necessarily the best one.3
6. Numerical example (nonzero debt)
The nonzero debt assumption affects the structure of the financial system in the
following way:
Applications Sources
ssss faiSS −++= − )1(1 | sss fDD −+= − )1()( 1 δδ
sss axxx −+= − )1()(IC)(IC 1 | ssss DSE −+= IC
(12)
where )(δsD is the debt at time s , δ is the interest rate on debt, Rf s ∈− is the instalment
for repaying the debt. The EVA and the SVA are computed as
)())((IC
))(IC()(ICEVA
11
1111
iDixx
DxiDxx
ss
sssss
−−−=−−−=
−−
−−−−
δδ
(13)
)-()(CISVA 11
11 −−
−− −−= ss
sss SSiDxx δ (14)
3 It may be shown that it is semantically ambiguous and seems to fall prey to some logical contradictions (see Magni (2001a)).
12
Eq. (13) coincides with eq. (1), where, obviously, x :=ROI.4 Eq. (14) is derived from eq. (7)
by subtracting interest on debt.
Suppose now that an investor has the opportunity of purchasing firm B at a price of 400. The
debt amounts to 600. Suppose that the initial investor’s endowment 0E is equal to 500 and
that the cost of capital is 13%. The return rate on the firm’s invested capital is a yearly 20%
and interest on debt is 15%. The decision maker extinguishes debt by paying off instalments
equal to 20 and 770.5 at time 1 and time 2 respectively, withdrawing the sums from the firm’s
Cash item. From the latter the investor also withdraws dividends to herself equal to 10 each
year up to the end of the third year, and invests them in account S. At the end of the fourth
year the firm will be liquidated and the terminal value is assumed to be 885.84. From a
financial perspective, the situation may be likened to a project partially financed by debt
Graphically,
Project B
0 1 2 3 4
|----------------------|-----------------------|-----------------------|-----------------------|
-1000 30 780.5 10 885.84
Debt
0 1 2 3 4
|----------------------|-----------------------|-----------------------|-----------------------|
+600 -20 -770.5 0 0
4 Eq. (13) coincides with the uncapitalized NPV periodic share in Peccati’s (1987, 1991, 1992) model.
13
Net Cash Flows
0 1 2 3 4
|-----------------------|----------------------|-----------------------|-----------------------|
-400 10 10 10 885.84
Net cash flows are the cash flows which are withdrawn from or reinvested in account S.
The computation of the EVAs and the SVAs is easy: it suffices to draw, for each period,
double-entry financial systems of the same type as in eq. (12), from which eqs. (13) e (14) are
derived. We have
38533.108784.885)13.1(3587.178
3587.17810)13.1(99.148
99.14810(1.13)123
12310 (1.13) 100S
100400500
4
3
2
1
0
=+==+=
=+==+=
=−=
S
S
S
S
084.885)2.1(2.738)(IC
2.73810)2.1(5.623)(IC
5.623105.770)2.1(1170)(IC
117010-20-(1.2)1000)(IC
1000)(IC
4
3
2
1
0
=−==−=
=−−===
=
x
x
x
x
x
2368.815)13.1(4485.721
4485.721(1.13)45.638
45.638(1.13) 565S
565)13.1(500
500
4
3
2
1
0
==
==
==
==
=
S
S
S
S
VFN1485.272
0898.543
46.489
442
400
44
33
22
11
00
−=−=−
=−
=−
=−
=−
SS
SS
SS
SS
SS
14
whence
6. Concluding remarks
This paper shows that the notion of residual income (excess profit) is conventional:
EVA is one amongst other possible ones. We have proposed an alternative model, the
Systemic Value Added (SVA), which is generated through appropriate considerations about
the diachronic evolution of the investor’s financial system. The choice of either model
depends on the piece of information the decision maker aims at drawing from the analysis.
0
0
05.770)15.1(670
67020)15.1(600
600
4
3
2
1
0
==
=−==−=
=
D
D
D
D
D
674.51)13.015.0(0)13.02.0(2.738EVA
645.43)13.015.0(0)13.02.0(5.623EVA
5.68)13.015.0(670)13.02.0(1170EVA
58)13.015.0(600)13.02.0(1000EVA
4
3
2
1
=−∗−−∗==−∗−−∗==−∗−−∗==−∗−−∗=
03833.770898.54313.0015.02.7382.0SVA
07.6146.48913.0015.05.6232.0SVA
04.7644213.067015.011702.0SVA
5840013.060015.010002.0SVA
4
3
2
1
=∗−∗−∗==∗−∗−∗==∗−∗−∗==∗−∗−∗=
NFV1485.272EVA)13.1(EVA)13.1(EVA)13.1(EVA 432
23
1 ==+++
NFV.1485.272SVASVASVASVA 4321 ==+++
15
The SVA is consistent with an “accounting” outlook of the investment, so to say, because it
may be seen as a difference between two profits derived from two double-entry sheets; one
relates to the alternative “invest in the project”, the other one relates to “reject the project”.
Essentially, for each of the two options, the future history of the financial system is described
ex ante, period after period. Then, the corresponding income are associated and compared,
period by period. The difference between the two alternative incomes is the Systemic Value
Added. Conversely, the EVA model is not concerned with the evolution of the financial
system: first, the capital invested at the beginning of each period is computed, and then
comparison is based on the idea that the capital invested can alternatively be invested either at
the rate x or at the rate i . The EVA model presupposes a comparison at equal invested
capital, whereas the SVA model implies that the capital invested is different, for the story of
the financial system in the two options is different.
The EVA model and the SVA model can be viewed as decomposition model of Net
Final Values. They conciliate from an aggregate perspective: the sum of the compounded
EVAs coincides with the sum of the uncompounded SVAs, which in turn coincides with the
Net Final Value. In this sense, we have presented a conciliation of accounting and finance:
EVA is grounded on elements typical of financial mathematics (just remind that the EVA
equals the periodic share of the NPV or NFV in Peccati’s model) so that one needs compound
(discount) residual incomes to get NFV (NPV), which is the global residual income referred
to the entire span of n periods. The SVA model is more akin to an accounting perspective,
where every fact is recorded in a double-entry sheet, that is, it is a system structured in various
items interacting in various ways. So doing, NFV is obtained as “crude” sum of all residual
incomes. This paper has then introduced a model that is, at the same time, accounting and
financial, because it follows a systemic approach to financially evaluate a project (or a firm).
16
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